Karp’s conjecture for independent sets in dense random graphs

Prove that for the Erdős–Rényi graph G(n, 1/2), no polynomial-time algorithm can, with high probability, compute an independent set of size at least (1+epsilon)·log_2 n for any fixed epsilon > 0.

Background

This classic conjecture by Karp concerns the dense regime (p = 1/2) and posits strong average-case hardness for computing unusually large independent sets in G(n, 1/2).

The paper references this conjecture to situate its results within the broader landscape of statistical–computational gaps for independent set problems across regimes.